Curvature, Conformal Mapping, and 2D Stationary Fluid Flows. Michael Taylor

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1 Curvature, Conformal Mapping, an 2D Stationary Flui Flows Michael Taylor 1. Introuction Let Ω be a 2D Riemannian manifol possibly with bounary). Assume Ω is oriente, with J enoting counterclockwise rotation by 90. As is well known, we can take a real-value function a stream function) ψ : Ω R assume ψ is constant on each component γ j of Ω) an then 1.1) v J ψ is a vector fiel on Ω tangent to Ω, if this is nonempty), efining a stationary solution to the Euler equation for incompressible invisci) flui flow, provie ψ has the property 1.2) ψ)z) ψz), z Ω. In particular, this hols provie ψ satisfies any nonlinear PDE of the form 1.3) ψ Φψ) on Ω, with ψ c j on γ j. Constructing such solutions provies many stationary invisci 2D flui flows. Remark: sometimes one sees the statement that 1.2) an 1.3) are equivalent, but actually 1.2) is more general than 1.3).) An example of 1.3) with particular geometrical import is 1.4) ψ K e 2ψ, with K a constant typically K ±1). The contact with geometry arises as follows. Suppose Ω has a flat metric e.g., Ω coul be a planar omain, or covere by a planar omain). Then multiplying its metric tensor by the factor e 2ψ prouces a metric of Gauss curvature K if an only if 1.4) hols. We will use this observation an conformal mapping techniques to construct solutions to 1.4), which then yiel interesting stationary solutions to the Euler equation, via 1.1). We note that the vorticity of such a flow is given by 1.5) ω ψ, so our solutions will typically have nontrivial vorticity. Thus the use mae here of conformal mapping contrasts with the well known use of conformal mappings to prouce irrotational stationary planar flows. 1

2 2 2. Cat s eye flows Our first example takes Ω C/Γ, where Γ {2πik : k Z}. We prouce a 2-parameter family of stationary flows on Ω, containing the 1-parameter family of Kelvin-Stuart cat s-eye flows given on p. 53 of [MB], following [S]. These solutions arise from 1.4) with K 1. We take the conformal iffeomorphism 2.1) E : Ω C \ {0}, Ez) e z, an pull back to Ω a metric of curvature 1 on C \ {0}. In fact, we take a family of metrics on C conformally equivalent to the stanar flat metric) of curvature 1, coming from conformal equivalence of C { } with S 2, the unit sphere in R 3, with its stanar metric of curvature 1). A conformal transformation of C { } to S 2 is given by 2.2) F : C S 2 where z x + iy C is mappe to u, v, t) S 2 R 3 u 2 + v 2 + t 2 1), via 2.3) u 2x x 2 + y 2 + 1, v 2y x 2 + y 2 + 1, t x2 + y 2 1 x 2 + y Uner this map, u 2 + v 2 + t 2 pulls back to 2.4) 4 x 2 + y 2 x 2 + y 2 + 1) 2 4 z 2 z 2 + 1) 2. If we pull back the metric 2.4) to Ω via 2.1) we get a conformal multiple e 2ψ of the stanar metric, yieling a stream function. We get a more general class of stream functions by making the following construction. Given ) a b 2.5) A Gl2, C), c the map 2.6) T A w) aw + b cw + z efines a conformal automorphism of C { }. The metric 2.4) pulls back to 2.7) 4 a bc 2 aw + b 2 + cw + 2 ) 2 w 2.

3 3 Via 2.1), this pulls back to the following metric on Ω: 2.8) 4 a bc 2 e z 2 ae z + b 2 + ce z + 2 ) 2 z 2 4 a bc 2 ae z/2 + be z/2 2 + ce z/2 + e z/2 2 ) 2 z 2. With z x + iy, note that 2.9) ae z/2 + be z/2 2 a 2 e x + b 2 e x + 2 Reab e iy ). Thus we can write the metric 2.8) on Ω as 2.10) 4 a bc 2 α 2 e x + β 2 e x + 2 Reα, β)e iy ) 2 x2 + y 2 ), where 2.11) α an ) ) a b, β C 2, c 2.12) α 2 a 2 + c 2, β 2 b 2 + 2, α, β) ab + c. The coefficient of x 2 + y 2 in 2.10) efines e 2ψ. An alternative formulation is 2.13) e ψ 1 2 et A 1 A e z/2 e z/2 ) 2 α 2 e x + β 2 e x + 2 Re α, β)e iy. 2 etα, β) ) a b This class of metrics is parametrize by A Gl2, C), but in fact c by reucing the numerator an enominator in 2.6) we can take A Sl2, C). Furthermore, multiplication of A on the left by a unitary matrix has a trivial effect, so the set of metrics in 2.10) or stream functions 2.13)) is effectively parametrize by SU2)\Sl2, C), equivalent to 3D hyperbolic space. To pick out the 1-parameter family of Kelvin-Stuart flows from 2.13), we specialize to ) ) a b 2.14) α, β R 2, α 2 β 2. c

4 4 Then 2.13) specializes to 2.15) e ψ α 2 cosh x + α β) cos y. α β If we set 2.16) γ α 2 α β, σ α β α β, then, given 2.14), we have 2.17) γ 1, σ ± γ 2 1. This recovers the Kelvin-Stuart stream function: 2.18) ψz) logγ cosh x + γ 2 1 cos y), γ 1. Back to the more general case 2.13), we can take 2.19) A Then 2.13) becomes with b b 1 + ib 2 ) ) 1 b, b C, > ) e ψ ex + b b )e x 2 + b 1 cos y + b 2 sin y, which reuces to 2.15) when b 2 0 an b with γ 1/). Note that if b b e iθ then Re be iy b cosy θ), so another way to write 2.20) is as 2.21) e ψ ex + b )e x 2 + b cosy θ). The phase θ is not physically significant, so we see we have a 2-parameter family of stream functions. We can just take b 0 in 2.19). We get γ 2.22) ψ log 2 ex + σ2 + 1 ) e x + σ cos y, 2γ where γ 1/ an σ b/ are inepenent positive numbers. The special case σ γ 2 1 γ 1) is 2.18).

5 Note that the argument of the logarithm in 2.22) has the form fx) + σ cos y. We see that f is convex, f± ) +, an 2.23) f x) 0 e 2x σ2 + 1 γ 2, f min σ Thus ψx, y) has local maxima where 2.23) hols an y 2kπ, k Z, an sale points where 2.23) hols an y 2kπ + π, k Z. The qualitative behavior of such stream functions an their associate flows are much the same as those of the special cases given by 2.18) The case K 1 Solutions to 1.4) with K 1 are also stream functions, yieling steay solutions to the Euler equations on omains Ω R 2. In particular, if R 2 \ Ω has at least two points, Ω has a canonical Poincaré metric, of the form e 2ψ x 2 + y 2 ), ψ satisfying 1.4) with K 1. Examples inclue 3.1) e 2ψ an 3.2) e 2ψ which is a limiting case of 4 1 z 2 ) 2, on D 1 {z R 2 : z < 1}, z log 1 ) 2, on D1 \ {0}, z 3.3) e 2ψ b 2, )) z sin on {z : e π/b b log 1 < z < 1}, z given b > 0. The formulas 3.2) an 3.3) can be erive from 3.1) via various conformal covering maps. See, e.g., the introuction to [MT].) These raial stream functions are not particularly interesting from the point of view of stuying flui flows, but one can pull back Poincaré metrics to other omains, via conformal mappings, an thereby obtain explicit formulas for other stream functions. In these cases, ψ blows up at Ω, but one can restrict to Ω a {z Ω : ψz) < a} to obtain smooth steay flows. For example, if we use the transformation 2.6) to pull back the metric tensor on D 1 \ {0} given by 3.2), we obtain a metric tensor e 2ψ z 2 with 3.4) e 2ψ a bc 2 az + b 2 cz + 2 log cz + az + b ) 2.

6 6 If we take 3.5) A ) 1 b, b < 1, b 1 then 3.4) is efine on D 1 \ { b} an also on {z : z > 1, z 1/b}). The attache Mathematica notebook Figure2A.nb) shows some streamlines when we take b 0.2 in 3.5). As one can see, the vector fiel v J ψ has a critical point of sale type at x 0.45, y 0, an a critical point of center type at x 0.25, y 0. These features are also present in the cat s eye flows, iscusse in 2. Compare Figure 2.4 in [MB]. If we pull back this metric on D 1 \ {0} to the upper half plane, minus 0, 1), via z z i)/z + i), we obtain the stream function given by 3.6) e 2ψ 4 z + i ) 2, z log z i whose streamlines are epicte in Figure2B.nb. Note the sale point at x 0, y In Figure3.nb we show streamlines of a stream function obtaine as follows. Pick b 0, 1) here b 0.15), take the Poincaré metric on D 1 \ { b} arising from 3.4) 3.5), an pull it back to D 1 \ {±i b} via the branche covering map z z 2. The resulting metric tensor is e 2ψ z 2 with 3.7) e 2ψ 41 b 2 ) z 2 z 2 + b 2 bz log bz2 + 1 z 2 + b ) 2. This metric tensor is egenerate at z 0. In fact we have ψz) as z 0, while ψz) + as z ±i b an as z 1. We can regar some close level curve of ψ aroun the origin as the bounary of an obstacle, an Figure3.nb shows streamlines of a flow in a region that is approximately a isk, with three obstacles. In Figure4.nb we show streamlines of a stream function obtaine as follows. Take the Poincaré metric on {z : Im z > 0, z i} given by 3.6) an pull it back to the strip {z : 0 < Im z < π}, with the point z πi/2 elete, via the map z e z. The resulting metric tensor is e 2ψ z 2 with 3.8) e 2ψ 1 cosh z 2 log ez + i e z i ) 2. This moels flow in a not quite straight) pipe, with an obstacle, the flow moving to the left at the top an to the right at the bottom. While we have welt on Poincaré metrics here with one sort of generalization for the case of D 1 \ {±i b}), more general metrics of the form e 2ψ z 2 with

7 curvature 1 escribe stream functions. We note that such a metric oes not exist on the space C/Γ iscusse in 2. Recall we ha K +1 there.) Inee, C/Γ is holomorphically covere by C, not by D 1, so it oes not have a Poincaré metric. Now if we ha any metric of the form e 2ψ z 2 on C/Γ with Gauss curvature K 1, then this woul provie a barrier function allowing the construction of a Poincaré metric cf. [MT], Proposition 7.1), giving a contraiction Applications to Beltrami flows A Beltrami flow on O R 3 is a vector fiel v on O tangent to the bounary, if nonempty), satisfying 4.1) curl vp) vp), p O. Such v is a steay solution to the 3D Euler equation. If ψ is a function on Ω R 2 satisfying 1.3), then the vector fiel on O Ω R efine by 4.2) v ψ y, ψ x, W ψ)) gives a Beltrami flow, provie W ψ) satisfies 4.3) W ψ)w ψ) Φψ); cf. [MB], As pointe out there, if ψ satisfies 1.4) with K +1, one can take W ψ) e ψ, so if ψ efines the cat s eye flow, i.e., if ψ is given by 2.18), or more generally if ψ is given by 2.22), then ψ y, ψ x, e ψ ) yiels a Beltrami flow. If ψ satisfies 1.4) with K 1, then 4.3) becomes 4.4) W ψ)w ψ) e 2ψ, which has solutions of the form 4.5) W ψ) A e 2ψ, for a given constant A > 0, as long as e 2ψ < A on Ω. This gives Beltrami flows on regions Ω R R 3 from stream functions constructe as inicate in 3. References [MB] A. Maja an A. Bertozzi, Vorticity an Incompressible Flow, Cambrige Univ. Press, [MT] R. Mazzeo an M. Taylor, Curvature an uniformization, Israel J. Math ), [S] J. Stuart, Stability problems in fluis, pp in Mathematical Problems in the Geophysical Sciences, AMS, Provience, R.I., 1971.